1. 3.1 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Elementary Statistics Eleventh Edition Chapter 3

2. 3.1 - 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots

3. 3.1 - 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Created by Tom Wegleitner, Centreville, Virginia Section 3-1 Review and Preview

4. 3.1 - 4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  Chapter 1 Distinguish between population and sample, parameter and statistic Good sampling methods: simple random sample, collect in appropriate ways  Chapter 2 Frequency distribution: summarizing data Graphs designed to help understand data Center, variation, distribution, outliers, changing characteristics over time Review

5. 3.1 - 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  Important Statistics Mean, median, standard deviation, variance  Understanding and Interpreting these important statistics Preview

6. 3.1 - 6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  Descriptive Statistics In this chapter we’ll learn to summarize or describe the important characteristics of a known set of data  Inferential Statistics In later chapters we’ll learn to use sample data to make inferences or generalizations about a population Preview

7. 3.1 - 7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 3-2 Measures of Center

8. 3.1 - 8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Measure of Center  Measure of Center the value at the center or middle of a data set

9. 3.1 - 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Arithmetic Mean  Arithmetic Mean (Mean) the measure of center obtained by adding the values and dividing the total by the number of values What most people call an average.

10. 3.1 - 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notation  denotes the sum of a set of values. x is the variable usually used to represent the individual data values. n represents the number of data values in a sample. N represents the number of data values in a population.

11. 3.1 - 11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Notation µ is pronounced ‘mu’ and denotes the mean of all values in a population x = n  x x is pronounced ‘x-bar’ and denotes the mean of a set of sample values x N µ =  x x

12. 3.1 - 12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  Advantages Is relatively reliable, means of samples drawn from the same population don’t vary as much as other measures of center Takes every data value into account Mean  Disadvantage Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center

13. 3.1 - 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Median  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘x-tilde’) ~  is not affected by an extreme value

14. 3.1 - 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Finding the Median 1.If the number of data values is odd, the median is the number located in the exact middle of the list. 2.If the number of data values is even, the median is found by computing the mean of the two middle numbers. First sort the values (arrange them in order), the follow one of these

15. 3.1 - 15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. 0.42 0.48 0.660.73 1.10 1.10 5.40 (in order - odd number of values) exact middle MEDIAN is 0.73 0.42 0.48 0.731.10 1.10 5.40 0.73 + 1.10 2 (in order - even number of values – no exact middle shared by two numbers) MEDIAN is 0.915

16. 3.1 - 16 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mode  Mode the value that occurs with the greatest frequency  Data set can have one, more than one, or no mode Mode is the only measure of central tendency that can be used with nominal data Bimodal two data values occur with the same greatest frequency Multimodalmore than two data values occur with the same greatest frequency No Modeno data value is repeated

17. 3.1 - 17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. a. 5.40 1.10 0.42 0.73 0.48 1.10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10 Mode - Examples  Mode is 1.10  Bimodal - 27 & 55  No Mode

18. 3.1 - 18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  Midrange the value midway between the maximum and minimum values in the original data set Definition Midrange = maximum value + minimum value 2

19. 3.1 - 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Carry one more decimal place than is present in the original set of values. Example: Calculate the mean of 2,3 & 5 Round-off Rule for Measures of Center

20. 3.1 - 20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Mean from a Frequency Distribution x = class midpoint f = frequency

21. 3.1 - 21 Copyright © 2004 Pearson Education, Inc. Mean from a Frequency Distribution AgeFrequency f Class Midpt. x f x 21-3028 31-4030 41-5012 51-602 61-702 71-802

22. 3.1 - 22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Weighted Mean x = w  (w x)  When data values are assigned different weights, we can compute a weighted mean.

23. 3.1 - 23 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Best Measure of Center

24. 3.1 - 24 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half  Skewed distribution of data is skewed if it is not symmetric and extends more to one side than the other Skewed and Symmetric

25. 3.1 - 25 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. The mean and median cannot always be used to identify the shape of the distribution. Shape of the Distribution

26. 3.1 - 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Skewness

27. 3.1 - 27 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Recap In this section we have discussed:  Types of measures of center Mean Median Mode  Mean from a frequency distribution  Weighted means  Best measures of center  Skewness

28. 3.1 - 28 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Section 3-3 Measures of Variation

29. 3.1 - 29 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Definition The range of a set of data values is the difference between the maximum data value and the minimum data value. Range = (maximum value) – (minimum value) It is very sensitive to extreme values; therefore not as useful as other measures of variation.

30. 3.1 - 30 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Round-Off Rule for Measures of Variation When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data. Round only the final answer, not values in the middle of a calculation.

31. 3.1 - 31 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Definition The standard deviation of a set of sample values, denoted by s, is a measure of variation of values about the mean.

32. 3.1 - 32 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Sample Standard Deviation Formula  ( x – x ) 2 n – 1 s =s =

33. 3.1 - 33 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Sample Standard Deviation (Shortcut Formula) n ( n – 1) s =s = n  x 2 ) – (  x ) 2

34. 3.1 - 34 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Sample Standard Deviation (Better Shortcut Formula)

35. 3.1 - 35 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Standard Deviation - Important Properties  The standard deviation is a measure of variation of all values from the mean.  The value of the standard deviation s is usually positive.  The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others).  The units of the standard deviation s are the same as the units of the original data values.

36. 3.1 - 36 Copyright © 2004 Pearson Education, Inc. Calculate the sample standard deviation of the number of students born in the 12 different months

37. 3.1 - 37 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Population Standard Deviation 2  ( x – µ ) N  = This formula is similar to the previous formula, but instead, the population mean and population size are used.

38. 3.1 - 38 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education  Population variance:  2 - Square of the population standard deviation  Variance  The variance of a set of values is a measure of variation equal to the square of the standard deviation.  Sample variance: s 2 - Square of the sample standard deviation s

39. 3.1 - 39 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Unbiased Estimator The sample variance s 2 is an unbiased estimator of the population variance  2, which means values of s 2 tend to target the value of  2 instead of systematically tending to overestimate or underestimate  2.

40. 3.1 - 40 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Variance - Notation s = sample standard deviation s 2 = sample variance  = population standard deviation  2 = population variance

41. 3.1 - 41 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Beyond the Basics of Measures of Variation Part 2

42. 3.1 - 42 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Range Rule of Thumb is based on the principle that for many data sets, the vast majority (such as 95%) of sample values lie within two standard deviations of the mean.

43. 3.1 - 43 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Range Rule of Thumb for Interpreting a Known Value of the Standard Deviation Informally define usual values in a data set to be those that are typical and not too extreme. Find rough estimates of the minimum and maximum “usual” sample values as follows: Minimum “usual” value (mean) – 2  (standard deviation) = Maximum “usual” value (mean) + 2  (standard deviation) =

44. 3.1 - 44 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Range Rule of Thumb for Estimating a Value of the Standard Deviation s To roughly estimate the standard deviation from a collection of known sample data use where range = (maximum value) – (minimum value) range 4 s s 

45. 3.1 - 45 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Properties of the Standard Deviation Measures the variation among data values Values close together have a small standard deviation, but values with much more variation have a larger standard deviation Has the same units of measurement as the original data

46. 3.1 - 46 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Properties of the Standard Deviation For many data sets, a value is unusual if it differs from the mean by more than two standard deviations Compare standard deviations of two different data sets only if the they use the same scale and units, and they have means that are approximately the same

47. 3.1 - 47 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education Empirical (or 68-95-99.7) Rule For data sets having a distribution that is approximately bell shaped, the following properties apply:  About 68% of all values fall within 1 standard deviation of the mean.  About 95% of all values fall within 2 standard deviations of the mean.  About 99.7% of all values fall within 3 standard deviations of the mean.

48. 3.1 - 48 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education The Empirical Rule

49. 3.1 - 49 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education The Empirical Rule

50. 3.1 - 50 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.Copyright © 2010 Pearson Education The Empirical Rule

51. 3.1 - 51 Copyright © 2004 Pearson Education, Inc. A survey of polar bears showed that they slept an average of 17.5 hours a day, with a standard deviation of 1.5 hours. Determine how many hours of sleep each of the following percentages received. Round all answers to the nearest tenth. 68 % of the polar bears? ______________ 95 % of the polar bears? ______________ 99.7 % of the polar bears? ______________

52. 3.1 - 52 Copyright © 2004 Pearson Education, Inc. On a standardized test, the mean score was 50 and the standard deviation was 3. What percent of the participants scored between 50 and 56? Between 44 and 47?

53. 3.1 - 53 Copyright © 2004 Pearson Education, Inc. The scores earned on a test have a mean of 80 and a standard deviation of 9. Approximately what percent of the class scored 71 % or higher?

54. 3.1 - 54 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 3-4 Measures of Relative Standing and Boxplots

55. 3.1 - 55 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Key Concept This section introduces measures of relative standing, which are numbers showing the location of data values relative to the other values within a data set. They can be used to compare values from different data sets, or to compare values within the same data set. The most important concept is the z score. We will also discuss percentiles and quartiles, as well as a new statistical graph called the boxplot.

56. 3.1 - 56 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  z Score (or standardized value) the number of standard deviations that a given value x is above or below the mean Z score

57. 3.1 - 57 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Sample Population x – µ z =  Round z scores to 2 decimal places Measures of Position z Score z = x – x s

58. 3.1 - 58 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: –2 ≤ z score ≤ 2 Unusual Values: z score 2

59. 3.1 - 59 Copyright © 2004 Pearson Education, Inc. Comparing Heights using z-scores With a height of 75 in., Lyndon Johnson was the tallest president of the past century. With a height of 85 in., Shaquille O’Neal is one of the tallest players in the NBA. Who is relatively taller? Lyndon Johnson among the presidents of the past century, or Shaquille O’Neal among the players in the NBA? Presidents of the past century have a mean height of 71.5 in. and a standard deviation of 2.1 inches. NBA players have a mean height of 80 inches and a standard deviation of 3.3 inches. Since the two men are from different populations, we must convert to z- scores to compare them.

60. 3.1 - 60 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Percentiles are measures of location. There are 99 percentiles denoted P 1, P 2,... P 99, which divide a set of data into 100 groups with about 1% of the values in each group.

61. 3.1 - 61 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Finding the Percentile of a Data Value Percentile of value x = 100 number of values less than x total number of values Round to the nearest whole number Example: Using the table below, find the percentile for the value 0f 29. 4.556.5720 29303540 415052606568 70 72747580100113116120125 132150160200225

62. 3.1 - 62 Copyright © 2004 Pearson Education, Inc. n total number of values in the data set k percentile being used L locator that gives the position of a value L = n k 100 Notation Converting from the k th Percentile to the Corresponding Data Value Example: Find the value of the 80 th percentile. 4.556.5720 29303540 415052606568 70 72747580100113116120125 132150160200225

63. 3.1 - 63 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Converting from the kth Percentile to the Corresponding Data Value

64. 3.1 - 64 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Quartiles  Q 1 (First Quartile) separates the bottom 25% of sorted values from the top 75%.  Q 2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%.  Q 3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%. Are measures of location, denoted Q 1, Q 2, and Q 3, which divide a set of data into four groups with about 25% of the values in each group.

65. 3.1 - 65 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Q 1, Q 2, Q 3 divide ranked scores into four equal parts Quartiles 25% Q3Q3 Q2Q2 Q1Q1 (minimum)(maximum) (median)

66. 3.1 - 66 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  Interquartile Range (or IQR): Q 3 – Q 1  10 - 90 Percentile Range: P 90 – P 10  Semi-interquartile Range: 2 Q 3 – Q 1  Midquartile: 2 Q 3 + Q 1 Some Other Statistics

67. 3.1 - 67 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  For a set of data, the 5-number summary consists of the minimum value the first quartile Q 1 the median (or Q 2 ) the third quartile, Q 3 and the maximum value 5-Number Summary

68. 3.1 - 68 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.  A boxplot (or box-and-whisker- diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q 1 ; the median; and the third quartile, Q 3. Boxplot

69. 3.1 - 69 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Boxplots Boxplot of Movie Budget Amounts

70. 3.1 - 70 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Boxplots - Normal Distribution Normal Distribution: Heights from a Simple Random Sample of Women

71. 3.1 - 71 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Boxplots - Skewed Distribution Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches

72. 3.1 - 72 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Outliers  An outlier is a value that lies very far away from the vast majority of the other values in a data set.

73. 3.1 - 73 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Important Principles  An outlier can have a dramatic effect on the mean.  An outlier can have a dramatic effect on the standard deviation.  An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.

74. 3.1 - 74 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Outliers for Modified Boxplots For purposes of constructing modified boxplots, we can consider outliers to be data values meeting specific criteria. In modified boxplots, a data value is an outlier if it is... above Q 3 by an amount greater than 1.5  IQR below Q 1 by an amount greater than 1.5  IQR or

75. 3.1 - 75 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Modified Boxplot Construction  A special symbol (such as an asterisk) is used to identify outliers.  The solid horizontal line extends only as far as the minimum data value that is not an outlier and the maximum data value that is not an outlier. A modified boxplot is constructed with these specifications:

76. 3.1 - 76 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Modified Boxplots - Example Pulse rates of females listed in Data Set 1 in Appendix B.

77. 3.1 - 77 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Make a modified boxplot of the following data. Use the formulas from the previous page to find outliers. 322,19,117,48,228,47,17,373,510,118,33,114,120,101,120,234